Variation of parameters - i have different particular soluti

[omar yahia]

i was trying to get a particular solution of a 3rd order ODE using the variation of parameters method
the homogeneous solution is y_h = c₁ e^-x + c₂ e^x + c₃ e^2x
the particular solution is y_p=y₁u₁+y₂u₂+y₃u₃
as u₁=∫ (w₁ g(x) /w) dx , u₂=∫ (w₂ g(x) /w) dx , u₃=∫ (w₃ g(x) /w) dx
w =
|y₁ y₂ y₃|
|y'₁ y'₂ y'₃|
|y''₁ y''₂ y''₃|

when i choose y₁ , y₂ , y₃ to be e^-x,e^x,e^2x i get an answer ,
but when i change the arrangement (like: e^x,e^-x,e^2x )
i get another different answer !

so , i have two questions
1 is it normal to have different results when changing who is y₁ , y₂ , y₃ , or am i doing something wrong?
2 if it is normal , does that mean i can have too many different particular solutions , just by changing who is y₁ , y₂ , y₃?

 

[mfb]

What are the different answers you get?
Maybe they are equivalent, and just look differently?

 

[omar yahia]

What are the different answers you get?
Maybe they are equivalent, and just look differently?

ummmmmm ... ahhhh .. actually ... when you asked me to show the results i went to prepare them for uploading
and as i was rewriting the solution i discovered an extra (2) multiplied in one little tiny term , will i fixed it and went on , and the results were the same indeed , i am terribly sorry for this mistake it happened because i trusted the solution of a senior without a thorough revision
thank you for your reply and i apologies again.

 

[HallsofIvy]

Never trust a senior!